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Chapter 25 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 25 Sigma (3) Notation The sign 3, called sigma, is a glorified symbol for addition that can be quite useful. It is utilized throughout mathematics, statistics, computer science and all other mathematical disciplines. With the 3 there is usually an index that typically is an i or j. Rather than define the 3 operation, I am going to try to teach it through a sequence of examples. Also, I will state some useful laws involving 3. 1 Chapter 25 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal General Laws G Exercise 1 Convince yourself that the four laws above are true. G Exercise 2 Prove that it is not true that: Three useful formulas that we proved in Section 6 (on induction) can be stated as: n ∑i = i =1 n ∑i 2 i =1 = n( n + 1) 2 n( n + 1)( 2 n + 1) 6 n 2 ( n + 1)2 i = ∑ 4 i =1 n 3 2 Chapter 25 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 1. None of the laws is difficult to prove. However, if you cannot do a proof, convince yourself through examples that all four laws are true. Note that the third law is a special case of the fourth law. 2. Almost any example will work. Try n = 2 with a1 = 1, a2 = 1, b1 = 2, b2 = 2. 3